Storage Size Determination Sfcr

8/14/2019 Storage Size Determination Sfcr

1/13

arXiv:1109

.4102v2

[math.OC

]10Jan2012

1

Storage Size Determination for Grid-Connected

Photovoltaic SystemsYu Ru, Jan Kleissl, and Sonia Martinez

AbstractIn this paper, we study the problem of determiningthe size of battery storage used in grid-connected photovoltaic(PV) systems. In our setting, electricity is generated from PVand is used to supply the demand from loads. Excess electricitygenerated from the PV can be stored in a battery to be used lateron, and electricity must be purchased from the electric grid if thePV generation and battery discharging cannot meet the demand.Due to the time-of-use electricity pricing, electricity can also bepurchased from the grid when the price is low, and be sold backto the grid when the price is high. The objective is to minimizethe cost associated with purchasing from (or selling back to) theelectric grid and the battery capacity loss while at the same timesatisfying the load and reducing the peak electricity purchasefrom the grid. Essentially, the objective function depends on thechosen battery size. We want to find a unique critical value(denoted as Ccref) of the battery size such that the total costremains the same if the battery size is larger than or equal toCcref, and the cost is strictly larger if the battery size is smallerthan Ccref. We obtain a criterion for evaluating the economicvalue of batteries compared to purchasing electricity from thegrid, propose lower and upper bounds on Ccref, and introducean efficient algorithm for calculating its value; these results arevalidated via simulations.

I. INTRODUCTION

The need to reduce greenhouse gas emissions due to fossil

fuels and the liberalization of the electricity market have led

to large scale development of renewable energy generatorsin electric grids [1]. Among renewable energy technologies

such as hydroelectric, photovoltaic (PV), wind, geothermal,

biomass, and tidal systems, grid-connected solar PV continued

to be the fastest growing power generation technology, with

a 70% increase in existing capacity to 13GW in 2008 [2].However, solar energy generation tends to be variable due to

the diurnal cycle of the solar geometry and clouds. Storage

devices (such as batteries, ultracapacitors, compressed air, and

pumped hydro storage [3]) can be used to i) smooth out the

fluctuation of the PV output fed into electric grids (capacity

firming) [2], [4], ii) discharge and augment the PV output

during times of peak energy usage (peak shaving) [5], or

iii) store energy for nighttime use, for example in zero-energybuildings and residential homes.

Depending on the specific application (whether it is off-

grid or grid-connected), battery storage size is determined

based on the battery specifications such as the battery storage

capacity (and the minimum battery charging/discharging time).

For off-grid applications, batteries have to fulfill the following

requirements: (i) the discharging rate has to be larger than

Yu Ru, Jan Kleissl, and Sonia Martinez are with the Mechani-cal and Aerospace Engineering Department, University of California,San Diego (e-mail: [email protected], [email protected],[email protected]).

or equal to the peak load capacity; (ii) the battery storage

capacity has to be large enough to supply the largest night time

energy use and to be able to supply energy during the longest

cloudy period (autonomy). The IEEE standard [6] provides

sizing recommendations for lead-acid batteries in stand-alone

PV systems. In [7], the solar panel size and the battery size

have been selected via simulations to optimize the operation of

a stand-alone PV system, which considers reliability measures

in terms of loss of load hours, the energy loss and the

total cost. In contrast, if the PV system is grid-connected,

autonomy is a secondary goal; instead, batteries can reduce

the fluctuation of PV output or provide economic benefits suchas demand charge reduction, and arbitrage. The work in [8]

analyzes the relation between available battery capacity and

output smoothing, and estimates the required battery capacity

using simulations. In addition, the battery sizing problem has

been studied for wind power applications [9][11] and hybrid

wind/solar power applications [12][14]. In [9], design of a

battery energy storage system is examined for the purpose

of attenuating the effects of unsteady input power from wind

farms, and solution to the problem via a computational proce-

dure results in the determination of the battery energy storage

systems capacity. Similarly, in [11], based on the statistics of

long-term wind speed data captured at the farm, a dispatch

strategy is proposed which allows the battery capacity to bedetermined so as to maximize a defined service lifetime/unit

cost index of the energy storage system; then a numerical

approach is used due to the lack of an explicit mathematical

expression to describe the lifetime as a function of the battery

capacity. In [10], sizing and control methodologies for a

zinc-bromine flow battery-based energy storage system are

proposed to minimize the cost of the energy storage system.

However, the sizing of the battery is significantly impacted by

specific control strategies. In [12], a methodology for calcu-

lating the optimum size of a battery bank and the PV array

for a stand-alone hybrid wind/PV system is developed, and a

simulation model is used to examine different combinations

of the number of PV modules and the number of batteries.In [13], an approach is proposed to help designers determine

the optimal design of a hybrid wind-solar power system;

the proposed analysis employs linear programming techniques

to minimize the average production cost of electricity while

meeting the load requirements in a reliable manner. In [ 14],

genetic algorithms are used to optimally size the hybrid system

components, i.e., to select the optimal wind turbine and PV

rated power, battery energy storage system nominal capacity

and inverter rating. The primary optimization objective is the

minimization of the levelized energy cost of the island system

over the entire lifetime of the project.
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In this paper, we study the problem of determining the

battery size for grid-connected PV systems. The targeted

applications are primarily electricity customers with PV arrays

behind the meter, such as residential and commercial build-

ings with rooftop PVs. In such applications, the objective is to

reduce the energy cost and the loss of investment on storage

devices due to aging effects while satisfying the loads and

reducing peak electricity purchase from the grid (instead of

smoothing out the fluctuation of the PV output fed into electric

grids). Our setting1 is shown in Fig.1. Electricity is generated

from PV panels, and is used to supply different types of

loads. Battery storage is used to either store excess electricity

generated from PV systems for later use when PV generation

is insufficient to serve the load, or purchase electricity from the

grid when the time-of-use pricing is lower and sell back to the

grid when the time-of-use pricing is higher. Without a battery,

if the load was too large to be supplied by PV generated

electricity, electricity would have to be purchased from the

grid to meet the demand. Naturally, given the high cost of

battery storage, the size of the battery storage should be chosen

such that the cost of electricity purchase from the grid and theloss of investment on batteries are minimized. Intuitively, if

the battery is too large, the electricity purchase cost could be

the same as the case with a relatively smaller battery. In this

paper, we show that there is a unique critical value (denoted

as Ccref, refer to Problem1) of the battery capacity such thatthe cost of electricity purchase and the loss of investments on

batteries remains the same if the battery size is larger than or

equal to Ccref and the cost is strictly larger if the battery sizeis smaller than Ccref. We obtain a criterion for evaluating theeconomic value of batteries compared to purchasing electricity

from the grid, propose lower and upper bounds on Ccref giventhe PV generation, loads, and the time period for minimizing

the costs, and introduce an efficient algorithm for calculatingthe critical battery capacity based on the bounds; these results

are validated via simulations.

The contributions of this work are the following: i) to the

best of our knowledge, this is the first attempt on determining

the battery size for grid-connected PV systems based on a

theoretical analysis on the lower and upper bounds of the

battery size; in contrast, most previous work are based on

simulations, e.g., the work in [8][12]; ii) a criterion for eval-

uating the economic value of batteries compared to purchasing

electricity from the grid is derived (refer to Proposition 4 and

Assumption2), which can be easily calculated and could be

potentially used for choosing appropriate battery technologies

for practical applications; and iii) lower and upper bounds onthe battery size are proposed, and an efficient algorithm is

introduced to calculate its value for the given PV generation

and dynamic loads; these results are then validated using sim-

ulations. Simulation results illustrate the benefits of employing

batteries in grid-connected PV systems via peak shaving and

cost reductions compared with the case without batteries (this

is discussed in SectionV.B).

1Note that solar panels and batteries both operate on DC, while the grid andloads operate on AC. Therefore, DC-to-AC and AC-to-DC power conversionis necessary when connecting solar panels and batteries with the grid andloads.

The paper is organized as follows. In the next section, we lay

out our setting, and formulate the storage size determination

problem. Lower and upper bounds on Ccref are proposedin Section III. Algorithms are introduced in Section IV to

calculate the value of the critical battery capacity. In SectionV,

we validate the results via simulations. Finally, conclusions

and future directions are given in SectionVI.

I I . PROBLEMF ORMULATION

In this section, we formulate the problem of determining

the storage size for a grid-connected PV system, as shown in

Fig.1. We first introduce different components in our setting.

A. Photovoltaic Generation

We use the following equation to calculate the electricity

generated from solar panels:

Ppv(t) =GHI(t) S , (1)

where

GHI (W m2

) is the global horizontal irradiation at thelocation of solar panels,

S (m2) is the total area of solar panels, and is the solar conversion efficiency of the PV cells.

The PV generation model is a simplified version of the one

used in [15] and does not account for PV panel temperature

effects.2

B. Electric Grid

Electricity can be purchased from or sold back to the grid.

For simplicity, we assume that the prices for sales and pur-

chases at timet are identical and are denoted as Cg(t)($/W h).

Time-of-use pricing is used (Cg(t) 0 depends ont) becausecommercial buildings with PV systems that would considera battery system usually pay time-of-use electricity rates.

In addition, with increased deployment of smart meters and

electric vehicles, some utility companies are moving towards

residential time-of-use pricing as well; for example, SDG&E

2Note that our analysis on determining battery capacity only relies onPpv(t) instead of detailed models of the PV generation. Therefore, morecomplicated PV generation models can also be incorporated into the costminimization problem as discussed in Section II.F.

Solar Panel

Electric Grid Battery

Load

DC/AC

DC/AC

AC/DC

AC Bus

Fig. 1. Grid-connected PV system with battery storage and loads.


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(San Diego Gas & Electric) has the peak, semipeak, offpeak

prices for a day in the summer season [16], as shown in Fig.3.

We use Pg(t)(W) to denote the electric power exchangedwith the grid with the interpretation that

Pg(t)> 0 if electric power is purchased from the grid, Pg(t)< 0 if electric power is sold back to the grid.

In this way, positive costs are associated with the electricity

purchase from the grid, and negative costs with the electricity

sold back to the grid. In this paper, peak shaving is enforced

through the constraint that

Pg(t) D ,

where D is a positive constant.

C. Battery

A battery has the following dynamic:

dEB(t)

dt =PB(t) , (2)

where EB(t)(W h) is the amount of electricity stored in thebattery at time t, and PB(t)(W) is the charging/discharging

rate; more specifically, PB(t)> 0 if the battery is charging, and PB(t)< 0 if the battery is discharging.

For certain types of batteries, higher order models exist (e.g.,

a third order model is proposed in [17], [18]).

To take into account battery aging, we useC(t)(W h)to de-note the usable battery capacity at time t. At the initial timet0,the usable battery capacity is Cref, i.e.,C(t0) = Cref 0. Thecumulative capacity loss at time t is denoted as C(t)(W h),and C(t0) = 0. Therefore,

C(t) = Cref C(t) .

The battery aging satisfies the following dynamic equation

dC(t)

dt =

ZPB(t) if PB(t)< 0

0 otherwise,(3)

whereZ >0 is a constant depending on battery technologies.This aging model is derived from the aging model in [5]

under certain reasonable assumptions; the detailed derivation

is provided in the Appendix. Note that there is a capacity loss

only when electricity is discharged from the battery. Therefore,

C(t) is a nonnegative and non-decreasing function oft.We consider the following constraints on the battery:

i) At any time, the battery chargeEB(t) should satisfy

0 EB(t) C(t) = Cref C(t) ,

ii) The battery charging/discharging rate should satisfy

PBmin PB(t) PBmax ,

where PBmin < 0, PBmin is the maximum batterydischarging rate, andPBmax> 0 is the maximum batterycharging rate. For simplicity, we assume that

PBmax= PBmin=C(t)

Tc=

Cref C(t)

Tc,

where constant Tc > 0 is the minimum time required tocharge the battery from0 to C(t)or discharge the batteryfrom C(t) to 0.

D. Load

Pload(t)(W) denotes the load at time t. We do not makeexplicit assumptions on the load considered in Section III

except that Pload(t) is a (piecewise) continuous function. Inresidential home settings, loads could have a fixed schedule

such as lights and TVs, or a relatively flexible schedule

such as refrigerators and air conditioners. For example, air

conditioners can be turned on and off with different schedules

as long as the room temperature is within a comfortable range.

E. Converters for PV and Battery

Note that the PV, battery, grid, and loads are all connected

to an AC bus. Since PV generation is operated on DC, a DC-

to-AC converter is necessary, and its efficiency is assumed to

be a constant pv satisfying

0< pv 1 .

Since the battery is also operated on DC, an AC-to-DC

converter is necessary when charging the battery, and a DC-

to-AC converter is necessary when discharging as shown in

Fig. 1. For simplicity, we assume that both converters have

the same constant conversion efficiency B satisfying

0< B 1 .

We define

PBC(t) =

BPB(t) if PB(t)< 0PB(t)B

otherwise,

In other words, PBC(t) is the power exchanged with the ACbus when the converters and the battery are treated as an entity.

Similarly, we can derive

PB(t) =

PBC(t)B

if PBC(t)< 0

BPBC(t) otherwise,

Note that2B is the round trip efficiency of the battery storage.

F. Cost Minimization

With all the components introduced earlier, now we can

formulate the following problem of minimizing the sum of

the net power purchase cost3 and the cost associated with the

battery capacity loss while guaranteeing that the demand from

3Note that the net power purchase cost include the positive cost to purchaseelectricity from the grid, and the negative cost to sell electricity back to thegrid.


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loads and the peak shaving requirement are satisfied:

minPB ,Pg

t0+Tt0

Cg()Pg()d+ KC(t0+ T)

s.t. pvPpv(t) + Pg(t) =PBC(t) + Pload(t) , (6)

dEB(t)

dt =PB(t) ,

dC(t)dt

=

ZPB(t) if PB(t)< 00 otherwise,

0 EB(t) Cref C(t) ,

EB(t0) = 0 , C(t0) = 0 ,

PBmin PB(t) PBmax ,

PBmax= PBmin=Cref C(t)

Tc,

PB(t) =

PBC(t)B

if PBC(t)< 0

BPBC(t) otherwise,

Pg(t) D , (7)

where t0 is the initial time, T is the time period consideredfor the cost minimization, K($/W h)> 0 is the unit cost forthe battery capacity loss. Note that:

i) No cost is associated with PV generation. In other words,

PV generated electricity is assumed free;

ii) KC(t0+ T) is the loss of the battery purchase invest-ment during the time period from t0 tot0+ Tdue to theuse of the battery to reduce the net power purchase cost;

iii) Eq. (6) is the power balance requirement for any time

t [t0, t0+ T];iv) the constraint Pg(t) D captures the peak shaving

requirement.

Given a battery of initial capacity Cref, on the one hand, if

the battery is rarely used, then the cost due to the capacityloss KC(t0 + T) is low while the net power purchase

costt0+Tt0

Cg()Pg()d is high; on the other hand, if thebattery is used very often, then the net power purchase costt0+Tt0

Cg()Pg()dis low while the cost due to the capacityloss KC(t0+ T) is high. Therefore, there is a tradeoff onthe use of the battery, which is characterized by calculating

an optimal control policy on PB(t), Pg(t) to the optimizationproblem in Eq. (7).

Remark 1 Besides the constraintPg(t) D, peak shaving isalso accomplished indirectly through dynamic pricing. Time-

of-use price margins and schedules are motivated by the

peak load magnitude and timing. Minimizing the net powerpurchase cost results in battery discharge and reduction in grid

purchase during peak times. If, however, the peak load for a

customer falls into the off-peak time period, then the constraint

onPg(t)limits the amount of electricity that can be purchased.Peak shaving capabilities of the constraint Pg(t) D anddynamic pricing will be illustrated in Section V.B.

G. Storage Size Determination

Based on Eq. (6), we obtain

Pg(t) = Pload(t) pvPpv(t) + PBC(t) .

Let u(t) = PBC(t), then the optimization problem in Eq. (7)can be rewritten as

minu

t0+Tt0

Cg()(Pload() pvPpv() + u())d+ KC(t0+ T)

s.t. dEB(t)

dt =

u(t)B

if u(t)< 0

Bu(t) otherwise,

dC(t)dt

=

Zu(t)

B if u(t)< 00 otherwise,

EB(t) 0 , EB(t0) = 0 , C(t0) = 0 ,

EB(t) + C(t) Cref ,

Bu(t)Tc+ C(t) Cref if u(t)> 0,

u(t)

BTc+ C(t) Cref if u(t)< 0,

Pload(t) pvPpv(t) + u(t) D . (8)

Now it is clear that only u(t) is an independent variable. Wedefine the set of feasible controls as controls that guarantee

all the constraints in the optimization problem in Eq. (8).

Let J denote the objective function

minut0+Tt0

Cg()(Pload() pvPpv() + u())d+ KC(t0+ T) .

If wefix the parameterst0, T , K , Z , T c, andD,J is a functionof Cref, which is denoted as J(Cref). If we increase Cref,intuitively J will decrease though may not strictly decrease(this is formally proved in Proposition2) because the battery

can be utilized to decrease the cost by

i) storing extra electricity generated from PV or purchasing

electricity from the grid when the time-of-use pricing is

low, and

ii) supplying the load or selling back when the time-of-usepricing is high.

Now we formulate the following storage size determination

problem.

Problem 1 (Storage Size Determination) Given the opti-

mization problem in Eq. (8) with fixed t0, T , K , Z , T c, andD, determine a critical value Ccref 0 such that

Cref< Ccref, J(Cref)> J(C

cref), and

Cref Ccref, J(Cref) = J(Ccref).

One approach to calculate the critical value Ccref is that wefirst obtain an explicit expression for the function J(Cref) by

solving the optimization problem in Eq. (8) and then solvefor Ccref based on the function J. However, the optimizationproblem in Eq. (8) is difficult to solve due to the nonlinear

constraints on u(t) and EB(t), and the fact that it is hard toobtain analytical expressions forPload(t) andPpv(t) in reality.Even though it might be possible to find the optimal control

using the minimum principle [19], it is still hard to get an

explicit expression for the cost function J. Instead, in the nextsection, we identify conditions under which the storage size

determination problem results in non-trivial solutions (namely,

Ccrefis positive and finite), and then propose lower and upperbounds on the critical battery capacity Ccref.


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III. BOUNDS ONCcref

Now we examine the cost minimization problem in Eq. (8).

SincePload(t) pvPpv(t) + u(t) D, or equivalently,u(t)D + pvPpv(t) Pload(t), is a constraint that has to be satisfiedfor any t [t0, t0 + T], there are scenarios in which eitherthere is no feasible control or u(t) = 0 for t [t0, t0+ T].Given Ppv(t), Pload(t), and D , we define

S1= {t [t0, t0+ T] | D+ pvPpv(t) Pload(t)< 0} , (9)

S2= {t [t0, t0+ T] | D+ pvPpv(t) Pload(t) = 0} , (10)

S3= {t [t0, t0+ T] | D+ pvPpv(t) Pload(t)> 0} . (11)

Note that4 S1S2 S3 = [t0, t0+T]. Intuitively,S1 is the setof time instants at which the battery can only be discharged,

S2 is the set of time instants at which the battery can bedischarged or is not used (i.e., u(t) = 0), and S3 is the set oftime instants at which the battery can be charged, discharged,

or is not used.

Proposition 1 Given the optimization problem in Eq. (8), if

i) t0 S1, orii) S3 is empty, oriii) t0 /S1 (or equivalently,t0 S2 S3), S3 is nonempty,

S1 is nonempty, and t1 S1, t3 S3, t1 < t3,

then either there is no feasible control or u(t) = 0 for t [t0, t0+ T].

Proof: Now we prove that, under these three cases, either

there is no feasible control or u(t) = 0 fort [t0, t0+ T].

i) Ift0 S1, then u(t0) D + pvPpv(t0) Pload(t0)< 0.However, since EB(t0) = 0, the battery cannot bedischarged at time t0. Therefore, there is no feasiblecontrol.

ii) IfS3 is empty, it means that u(t0) D +pvPpv(t0) Pload(t0) 0 for any t [t0, t0+ T], which implies thatthe battery can never be charged. IfS1 is nonempty, thenthere exists some time instant when the battery has to be

discharged. Since EB(t0) = 0 and the battery can neverbe charged, there is no feasible control. IfS1 is empty,thenu(t) = 0 for anyt [t0, t0+ T] is the only feasiblecontrol because EB(t0) = 0.

iii) In this case, the battery has to be discharged at time t1,but the charging can only happen at time instant t3 S3.Ift3 S3, t1 S1 such that t1 < t3, then the batteryis always discharged before possibly being charged. Since

EB(t0) = 0, then there is no feasible control.

Note that if the only feasible control is u(t) = 0 for t [t0, t0 + T], then the battery is not used. Therefore, we imposethe following assumption.

Assumption 1 In the optimization problem in Eq. (8), t0 S2 S3, S3 is nonempty, and either

S1 is empty, or S1 is nonempty, butt1 S1, t3 S3, t3< t1,

where S1, S2, S3 are defined in Eqs.(9), (10), (11).

4C= A B means C= A B andA B = .

Given Assumption 1, there exists at least one feasible

control. Now we examine how J(Cref) changes when Crefincreases.

Proposition 2 Consider the optimization problem in Eq. (8)

with fixed t0, T , K , Z , T c, and D. If C1ref < C

2ref, then

J(C1ref) J(C2ref).

Proof: Given C

1

ref, suppose control u

1

(t) achieves theminimum cost J(C1ref) and the corresponding states for thebattery charge and capacity loss areE1B(t)and C

1(t). Since

E1B(t) + C1(t) C1ref< C

2ref ,

Bu1(t)Tc+ C

1(t) C1ref< C2ref if u

1(t)> 0,

u1(t)

BTc+ C

1(t) C1ref< C2ref if u

1(t)< 0,

Pload(t) pvPpv(t) + u1(t) D ,

u1(t) is also a feasible control for problem (8) with C2ref, andresults in the cost J(C1ref). SinceJ(C

2ref) is the minimum cost

over the set of all feasible controls which include u1(t), wemust have J(C1ref) J(C

2ref).

In other words, J is non-increasing with respect to theparameter Cref, i.e., J is monotonically decreasing (thoughmay not be strictly monotonically decreasing). If Cref = 0,then0 C(t) Cref= 0, which implies that u(t) = 0. Inthis case, Jhas the largest value

Jmax= J(0) =

t0+Tt0

Cg()(Pload()pvPpv())d . (12)

Proposition 2 also justifies the storage size determination

problem. Note that the critical value Ccref (as defined in

Problem1)is unique as shown below.

Proposition 3 Given the optimization problem in Eq. (8) with

fixed t0, T , K , Z , T c, and D , Ccrefis unique.

Proof: We prove it via contradiction. Suppose Ccref is notunique. In other words, there are two different critical values

Cc1ref andCc2ref. Without loss of generality, suppose C

c1ref< C

c2ref.

By definition, J(Cc1ref) > J(Cc2ref) because C

c2ref is a critical

value, whileJ(Cc1ref) =J(Cc2ref)becauseC

c1refis a critical value.

A contradiction. Therefore, we must have Cc1ref=Cc2ref.

Intuitively, if the unit cost for the battery capacity loss K ishigher (compared with purchasing electricity from the grid),

then it might be preferable that the battery is not used at all,

which results in Ccref= 0, as shown below.

Proposition 4 Consider the optimization problem in Eq. (8)

with fixed t0, T , K , Z , T c, and D under Assumption1.

i) If

K (maxt Cg(t) mint Cg(t))B

Z ,

then J(Cref) = Jmax, which implies that Ccref= 0;ii) if

K < (maxt Cg(t) mint Cg(t))B

Z ,


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it is desirable to use battery storages. This criterion depends on

the pricing signal (especially the difference between the max-

imum and the minimum of the pricing signal), the conversion

efficiency of the battery converters, and the aging coefficient

Z.

Given the result in Proposition 4, we impose the following

additional assumption on the unit cost of the battery capacity

loss to guarantee that Ccref is positive.

Assumption 2 In the optimization problem in Eq. (8),

K 0 (because we requireCcref 0). Suppose

Ccref< Clbref=

TcB

(maxt

(Pload(t) pvPpv(t)) D),

or equivalently,

D


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Algorithm 1 Simple Algorithm for Calculating CcrefInput: The optimization problem in Eq. (8) with fixed

t0, T , K , Z , T c, D under Assumptions 1 and 2, the calculatedbounds Clbref, C

ubref, and parameters t,cap, cost

Output: An approximation ofCcref

1: Initialize Ciref= Cubref i cap, where i = 0, 1,...,L and L =

C

ubrefC

lbref

cap;

2: for i= 0, 1, ...,L do

3: Solve the optimization problem in Eq. (8) withCiref, and obtain

J(Ciref);4: if i 1 and J(Ciref) J(C

0ref) cost then

5: Set Ccref= Ci-1ref, and exit the for loop;

6: end if7: end for8: Output Ccref.

u(t)< 0. In addition, we use t as the sampling interval, anddiscretize Eqs. (2) and(3) as

EB(k+ 1) = EB(k) + PB(k)t ,

C(k+ 1) =C(k) ZPB(k)t if PB(k)< 0

C(k) otherwise .

With the indicator variable Iu(t) and the discretization ofcontinuous dynamics, the optimization problem in Eq. (8)

can be converted to a mixed integer programming problem

with indicator constraints (such constraints are introduced in

CPlex [20]), and can be solved using the CPlex solver [20]

to obtain J(Ciref). In the storage size determination problem,we need to check if J(Ciref) = J(C

cref), or equivalently,

J(Ciref) = J(Cubref); this is because J(C

cref) = J(C

ubref) due

to Ccref Cubref and the definition of C

cref. Due to numerical

issues in checking the equality, we introduce a small constant

cost > 0 so that we treat J(Ciref) the same as J(C

ubref) if

J(Ciref)J(Cubref)< cost. Similarly, we treat J(Ciref)> J(Cubref)ifJ(Ciref) J(C

ubref) cost. The detailed algorithm is given in

Algorithm1. At Step 4, ifJ(Ciref) J(C0ref) cost, or equiv-

alently, J(Ciref) J(Cubref) cost, we have J(C

iref)> J(C

ubref).

Because of the monotonicity property in Proposition 2, we

know J(Cjref) J(Ciref) > J(C

i-1ref) = J(C

ubref) for any

j = i+ 1,...,L. Therefore, the for loop can be terminated,and the approximated critical battery capacity is Ci-1ref.

It can be verified that Algorithm 1 stops after at most L + 1steps, or equivalently, after solving at most

Cubref C

lbref

cap + 1

optimization problems in Eq. (8). The accuracy of the critical

battery capacity is controlled by the parameters t, cap, cost.Fixingt, cost, the output is within

[Ccref cap, Ccref+ cap] .

Therefore, by decreasingcap, the critical battery capacity canbe approximated with an arbitrarily prescribed precision.

Since the function J(Cref) is a non-increasing function ofCref, we propose Algorithm 2 based on the idea of bisectionalgorithms. More specifically, we maintain three variables

C1ref< C3ref< C

2ref ,

Algorithm 2 Efficient Algorithm for Calculating CcrefInput: The optimization problem in Eq. (8) with fixed

t0, T , K , Z , T c, D under Assumptions 1 and 2, the calculatedbounds Clbref, C

ubref, and parameters t, cap, cost

Output: An approximation ofCcref

1: Let C1ref= Clbref and C

2ref= C

ubref;

2: Solve the optimization problem in Eq. (8) with C2ref, and obtainJ(C2ref);

3: Let sign = 1;4: while sign = 1 do

5: Let C3ref= C

1ref+C

2ref

2 ;

6: Solve the optimization problem in Eq. (8) withC3ref, and obtainJ(C3ref);

7: ifJ(C3ref) J(C2ref)< cost then

8: Set C2ref= C3ref, and J(C

2ref) = J(C

ubref);

9: else10: Set C1ref= C

3ref and set J(C

1ref) with J(C

3ref);

11: ifC2ref C3ref< cap then

12: Set sign = 0;13: end if14: end if15: end while16: Output Ccref= C

2ref.

in which C1ref (or C2ref) is initialized as C

lbref (or C

ubref). We set

C3ref to be C1ref+C

2ref

2 . Due to Proposition2, we have

J(C1ref) J(C3ref) J(C

2ref) .

IfJ(C3ref) = J(C2ref) (or equivalently, J(C

3ref) =J(C

ubref); this

is examined in Step 7), then we know that C1ref Ccref C

3ref;

therefore, we update C2ref with C3ref but do not update

7 the

value J(C2ref). On the other hand, ifJ(C3ref) > J(C

2ref), then

we know that C2ref Ccref C

3ref; therefore, we update C

1ref

with C3ref and set J(C1ref) with J(C

3ref). In this case, we also

check ifC

2

ref C

3

ref< cap: if it is, then output C

2

ref since weknow the critical battery capacity is between C3ref and C2ref;

otherwise, the while loop is repeated. Since every execution

of the while loop halves the interval [C1ref, C2ref] starting from

[Clbref, Cubref], the maximum number of executions of the while

loop is log2CubrefC

lbref

cap, and the algorithm requires solving at

most

log2Cubref C

lbref

cap + 1

optimization problems in Eq. (8). This is in contrast to

solvingCubrefC

lbref

cap + 1 optimization problems in Eq. (8) using

Algorithm1.

Remark 4 Note that the critical battery capacity can beimplemented by connecting batteries with fixed capacity in

parallel because we only assume that the minimum battery

charging time is fixed.

V. SIMULATIONS

In this section, we calculate the critical battery capacity

using Algorithm 2, and verify the results in Section III via

simulations. The parameters used in Section II are chosen

7Note that if we update J(C2ref) withJ(C3ref

), then the difference betweenJ(C2

ref) and J(Cub

ref) can be amplified when C2

ref is updated again later on.


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9

0

200

400

600

800

1000

1200

1400

1600

1800

Jul 8, 2010 Jul 9, 2010 Jul 10, 2010 Jul 11, 2010

Ppv

(w)

(a) Starting from July 8, 2011.

0

500

1000

1500

Jul 13, 2010 Jul 14, 2010 Jul 15, 2010 Jul 16, 2010

Ppv

(w)

(b) Starting from July 13, 2011.

Fig. 2. PV output based on GHI measurement at La Jolla, California, wheretick marks indicate noon local standard time for each day.

based on typical residential home settings and commercial

buildings.

A. Setting

The GHI data is the measured GHI in July 2010 at La

Jolla, California. In our simulations, we use = 0.15, andS = 10m2. Thus Ppv(t) = 1.5 GHI(t)(W). We have twochoices for t0:

t0 is 0000 h local standard time (LST) on Jul 8, 2010,and the hourly PV output is given in Fig. 2(a) for the

following four days starting from t0 (this corresponds tothe scenario in which there are relatively large variations

in the PV output);

t0 is 0000 h LST on Jul 13, 2010, and the hourly PVoutput is given in Fig. 2(b) for the following four days

starting fromt0 (this corresponds to the scenario in whichthere are relatively small variations in the PV output).

The time-of-use electricity purchase rate Cg(t) is

16.5/kW h from 11AM to 6PM (on-peak); 7.8/kW hfrom 6AM to 11AM and 6PM to 10PM (semi-

peak);

6.1/kW h for all other hours (off-peak).

This rate is for the summer season proposed by SDG&E [16],

and is plotted in Fig.3.

Since the battery dynamics and aging are characterized by

continuous ordinary differential equations, we use t= 1h as

0 1 2 3 4 5 6 7 8 9 10 1 1 12 1 3 14 1 5 1 6 17 1 8 19 2 0 21 2 2 23 2 424

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

4

Hours of a day

TImeofuserate($\Wh)

Fig. 3. Time-of-use pricing for the summer season at San Diego, CA [16].

the sampling interval, and discretize Eqs. (2) and (3) as

EB(k+ 1) = EB(k) + PB(k) ,

C(k+ 1) =

C(k) ZPB(k) if PB(k)< 0

C(k) otherwise .

In simulations, we assume that lead-acid batteries are used;

therefore, the aging coefficient is Z= 3 104

[5], the unitcost for capacity loss is K = 0.15$/W h based on the costof150$/kWh [21], and the minimum charging time is Tc =12h [22].

For the PV DC-to-AC converter and the battery DC-to-

AC/AC-to-DC converters, we use pv = B = 0.9. It can beverified that Assumption2holds because the threshold value

forK is8

(maxt Cg(t) mint Cg(t))BZ

=

(16.5 6.1) 105 0.9

3 104 0.3120 .

For the load, we consider two typical load profiles: the res-

idential load profile as given in Fig. 4(a), and the commercial

load profile as given in Fig. 4(b). Both profiles resemble the

corresponding load profiles in Fig. 8 of [15].9 Note that in

the residential load profile, one load peak appears in the early

morning, and the other in the late evening; in contrast, in the

commercial load profile, the two load peaks appear during

the daytime and occur close to each other. For multiple day

simulations, the load is periodic based on the load profiles in

Fig.4.

For the parameter D, we use D = 800(W). Since themaximum of the loads in Fig. 4is around 1000(W), we willillustrate the peak shaving capability of battery storage. It can

be verified that Assumption 1holds.

B. Results

We first examine the storage size determination problem

using both Algorithm 1 and Algorithm 2 in the following

setting (called the basic setting):

8Suppose the battery is a Li-ion battery with the same aging coefficient asa lead-acid battery. Since the unit cost K = 1.333$/Wh based on the costof1333$/kWh [21] (and K = 0.78$/Wh based on the 10-year projectedcost of780$/kWh [21]), the use of such a battery is not as competitive asdirectly purchasing electricity from the grid.

9However, simulations in [15] start at 7AM so Fig. 4 is a shifted versionof the load profile in Fig. 8 in [15].


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10

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24100

200

300

400

500

600

700

800

900

1000

Hours of a day

Load(w)

(a) Residential load averaged at 536.8(W).

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24100

200

300

400

500

600

700

800

900

1000

1100

Hours of a day

Load(w)

(b) Commercial load averaged at 485.6(W).

Fig. 4. Typical residential and commercial load profiles.

the cost minimization duration is T= 24(h), t0 is 0000 h LST on Jul 13, 2010, and the load is the residential load as shown in Fig. 4(a).

The lower and upper bounds in Proposition5are calculated as

2667(W h) and 39269(W h). When applying Algorithm1, we

choosecap = 10(W h) and cost= 104

, and haveL = 3660.When running the algorithm, 2327 optimization problems inEq. (8) have been solved. The maximum cost Jmax is0.1921while the minimum cost is 0.3222, which is larger than thelower bound 2.5483 as calculated based on Proposition 4.The critical battery capacity is calculated to be 16089(W h).In contrast, only

log2Cubrefcap

+ 1 = 13

optimization problems in Eq.(8) are solved using Algorithm2

while obtaining the same critical battery capacity.

Now we examine the solution to the optimization problem

in Eq. (8) in the basic setting with the critical battery capacity16089(W h). We solve the mixed integer programming prob-lem using the CPlex solver [20], and the objective function is

J(16089) = 0.3222. PB(t), EB(t) and C(t) are plotted inFig.5(a). The plot ofC(t)is consistent with the fact that thereis capacity loss (i.e., the battery ages) only when the battery

is discharged. The capacity loss is around 2.2(W h), and

C(t0+ T)

Cref=

2.2

16089= 1.4 104 0 ,

which justifies the assumption we make when linearizing the

nonlinear battery aging model in the Appendix. The dynamic

0 5 10 15 202000

0

2000

PB(W): Battery Charging/Discharging Profile

0 5 10 15 200

5000

10000

EB(Wh): Battery Charge Profile

0 5 10 15 201.6086

1.6087

1.6088

1.6089x 10

4 C(Wh): Capacity

(a)

0 5 10 15 200.5

1

1.5

2x 10

4C

g($/Wh): Dynamic Pricing

0 5 10 15 202000

0

2000


0 5 10 15 204000

2000

0

2000

Pload

(W): Load (solid red curve), and Pg(W): Net Power Purchase (dotted blue curve)

(b)

17 18 19 20 21 22 231000

800

600

400

200

0

200

400

600

800

1000

Pload


(c)

Fig. 5. Solution to a typical setting in which t0 is on Jul 13, 2010, T =24(h), the load is shown in Fig. 4(a), and Cref= 16089(Wh).

pricing signal Cg(t), PB(t), Pload(t) and Pg(t) are plotted inFig.5(b). From the second plot in Fig.5(b), it can be observed

that the battery is charged when the time-of-use pricing is low

in the early morning, and is discharged when the time-of-usepricing is high. From the third plot in Fig. 5(b), it can be

verified that, to minimize the cost, electricity is purchased

from the grid when the time-of-use pricing is low, and is

sold back to the grid when the time-of-use pricing is high;

in addition, the peak demand in the late evening (that exceeds

D= 800(W)) is shaved via battery discharging, as shown indetail in Fig. 5(c).

These observations also hold forT= 48(h). In this case, wechangeTto be48(h)in the basic setting, and solve the batterysizing problem. The critical battery capacity is calculated to be

Ccref= 16096(W h). Now we examine the solution to the opti-


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11

0 5 10 15 20 25 30 35 40 452000

0

2000


0 5 10 15 20 25 30 35 40 450

5000

10000

EB(W): Battery Charge Profile

0 5 10 15 20 25 30 35 40 451.609

1.6092

1.6094

1.6096x 10

4 C(Wh): Capacity

(a)

0 5 10 15 20 25 30 35 40 450.5

1

1.5

2x 10

4C


0 5 10 15 20 25 30 35 40 452000

0

2000


0 5 10 15 20 25 30 35 40 454000

2000

0

2000

Pload


(b)

Fig. 6. Solution to a typical setting in which t0 is on Jul 13, 2010, T =48(h), the load is shown in Fig. 4(a), and Cref= 16096(Wh).

mization problem in Eq.(8) with the critical battery capacity

16096(W h), and obtainJ(16096) =0.6032.PB(t),EB(t),C(t) are plotted in Fig. 6(a), and the dynamic pricing signalCg(t),PB(t),Pload(t) and Pg(t) are plotted in Fig.6(b). Notethat the battery is gradually charged in the first half of each

day, and then gradually discharged in the second half to be

empty at the end of each day, as shown in the second plot of

Fig.6(a).

To illustrate the peak shaving capability of the dynamic

pricing signal as discussed in Remark1, we change the load

to the commercial load as shown in Fig. 4(b) in the basic

setting, and solve the battery sizing problem. The critical

battery capacity is calculated to be Ccref = 13352(W h), andJ(13352) = 0.1785. The dynamic pricing signal Cg(t),PB(t), Pload(t) and Pg(t) are plotted in Fig. 7. For thecommercial load, the duration of the peak loads coincides

with that of the high price. To minimize the total cost, duringpeak times the battery is discharged, and the surplus electricity

from PV after supplying the peak loads is sold back to the

grid resulting in a negative net power purchase from the grid,

as shown in the third plot in Fig. 7. Therefore, unlike the

residential case, the high price indirectly forces the shaving of

the peak loads.

Now we consider settings in which the load could be either

residential loads or commercial loads, the starting time could

be on either Jul 8 or Jul 13, 2010, and the cost optimization

duration can be24(h), 48(h), 96(h). The results are shown inTables I and II. In Table I, t0 is on Jul 8, 2010, while in

0 5 10 15 200.5

1

1.5

2x 10

4C


0 5 10 15 204000

2000

0

2000


0 5 10 15 204000

2000

0

2000

Pload


Fig. 7. Solution to a typical setting in which t0 is on Jul 13, 2010, T =24(h), the load is shown in Fig. 4(b), and Cref= 13352(Wh).

Table II, t0 is on Jul 13, 2010. In the pair (24, R), 24 refersto the cost optimization duration, andR stands for residentialloads; in the pair(24, C), C stands for commercial loads.

We first focus on the effects of load types. From Tables I

and II, the commercial load tends to result in a higher cost

(even though the average cost of the commercial load issmaller than that of the residential load) because the peaks of

the commercial load coincide with the high price. When there

is a relatively large variation of PV generation, commercial

loads tend to result in larger optimum battery capacity Ccref asshown in Table I, presumably because the peak load occurs

during the peak pricing period and reductions in PV production

have to be balanced by additional battery capacity; when there

is a relatively small variation of PV generation, residential

loads tend to result in larger optimum battery capacity Ccref asshown in TableII. If t0 is on Jul 8, 2010, the PV generationis relatively lower than the scenario in which t0 is on Jul 13,2010, and as a result, the battery capacity is smaller and the

cost is higher. This is because it is more profitable to store

PV generated electricity than grid purchased electricity. From

TablesI andII, it can be observed that the cost optimization

duration has relatively larger impact on the battery capacity

for residential loads, and relatively less impact for commercial

loads.

In TablesI andII, the row Jmax corresponds to the cost inthe scenario without batteries, the row J(Ccref) corresponds tothe cost in the scenario with batteries of capacity Ccref, and therowSavings10 corresponds to Jmax J(Ccref). For TableI,wecan also calculate the relative percentage of savings using the

formula JmaxJ(C

cref)

Jmax, and get the row Percentage. One obser-

vation is that the relative savings by using batteries increase asthe cost optimization duration increases. For example, when

T= 96(h)and the load type is residential,16.10%cost can besaved when a battery of capacity 18320(W h) is used; whenT = 96(h) and the load type is commercial, 25.79% costcan be saved when a battery of capacity 15747(W h) is used.This clearly shows the benefits of utilizing batteries in grid-

connected PV systems. In Table II,only the absolute savings

are shown since negative costs are involved.

10Note that in Table II, part of the costs are negative. Therefore, the wordEarnings might be more appropriate than Savings.


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[16] SDG&E proposal to charge electric rates for different times

of the day: The devil lurks in the details. [Online]. Available:http://www.ucan.org/energy/electricity/sdge proposal charge electric rates different times day devil lurks details

[17] M. Ceraolo, New dynamical models of lead-acid batteries, IeeeTransactions On Power Systems, vol. 15, pp. 11841190, 2000.

[18] S. Barsali and M. Ceraolo, Dynamical models of lead-acid batteries:Implementation issues, IEEE Transactions On Energy Conversion,vol. 17, pp. 1623, Mar. 2002.

[19] A. E. Bryson and Y.-C. Ho, Applied Optimal Control: Optimization,Estimation and Control. New York, USA: Taylor & Francis, 1975.

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[21] D. Ton, G. H. Peek, C. Hanley, and J. Boyes, Solar energy gridintegration systems energy storage, Sandia National Lab, Tech. Rep.,2008.

[22] Charging information for lead acid batteries. [Online]. Available:http://batteryuniversity.com/learn/article/charging the lead acid battery
http://www.ucan.org/energy/electricity/sdge_proposal_charge_electric_rates_different_times_day_devil_lurks_detailshttp://batteryuniversity.com/learn/article/charging_the_lead_acid_batteryhttp://www-01.ibm.com/software/integration/optimization/cplex-optimizerhttp://batteryuniversity.com/learn/article/charging_the_lead_acid_batteryhttp://batteryuniversity.com/learn/article/charging_the_lead_acid_batteryhttp://www-01.ibm.com/software/integration/optimization/cplex-optimizerhttp://www.ucan.org/energy/electricity/sdge_proposal_charge_electric_rates_different_times_day_devil_lurks_details

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